i6 Algebra. 



4. In the case just considered we have the same poiver, but a 

 and b may be different. In Art. 2 we had the same letter, but the 

 powers may have been different. The student should net con- 

 fuse these two cases. 



5. The equation a"a''—a"^'' may be extended. Multiplying 

 both sides of the equation by a^ we have 



.*. a"a''a^=a"^''a^. 

 By equation {a) the right hand member equals a"+'+^. 



and so on, evidently, for any number of factors. 

 Now the exponents n, r, p may all be the same. 



.-. a"a"=^d"\ or f «"/=«-", 

 and a"a"a"=a^'\ or (a")^=a^", 

 and so on. Therefore, evidently, 



(a")'=a'"\ {b) 



11 and r being any positive ivhole numbers. This formula will be 

 referred to as formula {b). 



6. The equation a"b"=(ab)" may be extended. Multiplying 

 both sides of this equation by c" we obtain 



a"b"c"= (ab)"c "= (abc)", 

 and so on, evidently, for any number of factors. Hence, the 

 product of the n\.h. powers of any number of quantities is equal to 

 the nt\i power of the product of those quantities. 



KXAMPIvES. 

 Multiply .1-5 by x\ 

 Multiply x'' by x^. 

 Multiply x^ by x^. 

 Multiply —x^ by x^. 

 Multiply —x"^ by — x^ 

 Multiply (—x)' by (—x)\ 

 Multiply x^hy (—x)\ 

 Multiply x^ by (—x)\ 

 9. Multiply ri/ by ri/. 



10. Multiply r-i/ by (-^)\ 



11. Multiply (2a)^ by a\ 



12. Multiply 2^ a'^ and a^ together. 

 ij. Multiply (2a)^ by a^. 



